I would like to know how one could show and prove that a given motion is simple harmonic motion.
Once given an answer, I'll apply that technique to an example I am trying to figure out.
Thank you in advance!
I believe a motion can be proved simple harmonic, if the relation between its is as such:
$$
a_x = – \omega^2\cdot x
$$
And as such the period time is:
$$
T =\frac{2\pi}{\omega}
$$
Question-so-far: How do you prove such for a given force $F = \frac{G\cdot m_e \cdot M}{R_E} \cdot r$ ? Or any force that has non-trivial constants?
Best Answer
If the total force $F$ on a mass $m$ follows Hooke's law,
$$F~=~-kx,$$
then one can use Newton's 2nd law
$$F=ma,$$
to infer that the motion is a simple harmonic motion
$$ a =-\omega^2x, \qquad\qquad \frac{2\pi}{T}~=~\omega~=~ \sqrt{\frac{k}{m}}~,$$
cf. OP's correct belief. Now it only remains to solve the ODE
$$ \frac{d^2x(t)}{dt^2}~=~-\omega^2x(t), $$
which is a pure math exercise.